Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-18T04:25:53.420Z Has data issue: false hasContentIssue false
  • English
  • Français

Permanence and asymptotic stability in diagonally convex reaction–diffusion systems

Published online by Cambridge University Press:  14 November 2011

Jan H. van Vuuren  and
John Norbury
Jan H. van Vuuren
Affiliation:
Department of Applied Mathematics, University of Stellenbosch, Private Bag X1, Matieland, 7602, South Africa e-mail: vuuren@maties.sun.ac.za
John Norbury
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles, Oxford OX1 3LB, UK e-mail: ecmigb@vax.ox.ac.uk
Get access Rights & Permissions [Opens in a new window]

Extract

Reaction–diffusion systems are widely used to model competition in, for example, the scientific fields of biology, chemistry, medicine and industry. It is often not too difficult to establish positive uniform upper bounds on solution components of such systems, but the task of establishing strictly positive uniform lower bounds (when they exist) can be quite troublesome. Conditions for the existence of such lower bounds in Lotka–Volterra competitve models are already well known. A general context for the understanding of these conditions is provided in this paper, by establishing more general permanence criteria (for the nonexplosion and noncollapse to zero of solutions) in the class of diagonally convex competitive reaction–diffusion systems with zero flux Neumann boundary conditions. This class admits most famous competition models as special cases. The asymptotic (large time) behaviour of positive solutions within the bounds of permanence is also considered and it is shown that the methods of proof for asymptotic stability (and hence resilience) normally associated with order-preserving systems (such as comparison arguments) are also applicable, in a slightly generalised form, to competitive systems as long as the competitive interactions are not too strong. The general criteria for permanence obtained here provide a natural method for developing new and easily verifiable permanence conditions for a host of non Lotka–Volterra competition models, as is illustrated by considering three famous special cases. In one of these cases known results are recovered, while in the other two cases new conditions for solution permanence are established.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 1 , 1998 , pp. 147 - 172
Copyright
Copyright © Royal Society of Edinburgh 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Ahmad, S. and Lazer, A. C.. Asymptotic behaviour of solutions of periodic competition diffusion systems. Nonlinear Anal. 13 (1989), 263–84.CrossRefGoogle Scholar
2Alvarez, C. and Lazer, A. C.. An application of topological degree to the periodic competing species problem. J. Austral. Math. Soc. Ser. B 28 (1986), 202–19.CrossRefGoogle Scholar
3Ayala, F. J., Gilpin, M. E. and Ehrenfeld, J. G.. Competition between species: theoretical models and experimental tests. Theoret. Population Biol. 4 (1973), 331–56.CrossRefGoogle ScholarPubMed
4Brown, P. N.. Decay to uniform states in ecological interactions. SIAM J. Appl. Math. 38(1) (1980), 22–37.CrossRefGoogle Scholar
5Burton, T. A. and Hutson, V.. Permanence for non-autonomous predator–prey systems. Differential Integral Equations 4 (1991), 1269–80.CrossRefGoogle Scholar
6Cartrell, R. S., Cosner, C. and Hutson, V.. Permanence in ecological systems with spatial heterogeneity. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 533–59.CrossRefGoogle Scholar
7Chen, N. and Lu, Z. Y.. Global stability of periodic solutions to reaction–diffusion systems. Math. Appl. 4(1991), 111–14.Google Scholar
8Cosner, C. and Lazer, A. C.. Stable coexistence states in the Lotka–Volterra competition model with diffusion. SIAM J. Appl. Math. 44 (1984), 1112–32.CrossRefGoogle Scholar
9Mottoni, P. De and Schiaffino, A.. Competition systems with periodic coefficients: a geometrical approach. J. Math. Biol. 11 (1981), 319–35.CrossRefGoogle Scholar
10Fan, G. and Leung, A.. Existence and stability of periodic solutions for competing species diffusion systems with Dirichlet boundary conditions. Appl. Anal. 39 (1990), 119–49.CrossRefGoogle Scholar
11Fernands, M. L. C. and Zanolin, F.. Repelling conditions for boundary sets using Liapunov-like functions. II: Persistence and periodic solutions. J. Differential Equations 86 (1990), 33–58.CrossRefGoogle Scholar
12Fonda, A.. Uniformly persistent semi-dynamical systems. Proc. Amer. Math. Soc. 104 (1988), 111–16.CrossRefGoogle Scholar
13Gilpin, M. E. and Ayala, F. J.. Global models of growth and competition. Proc. Nat. Acad. Sci. USA 70(1973), 3590–3.CrossRefGoogle ScholarPubMed
14Gilpin, M. E. and Ayala, F. J.. Schoener's models and Drosophila competition. Theoret. Population Biol. 9(1974), 12–14.CrossRefGoogle Scholar
15Gopalsamy, K.. Global stability in a periodic Lotka–Volterra system. J. Austral. Math. Soc. Ser. B 27(1985), 66–72.CrossRefGoogle Scholar
16Hale, J. K. and Waltman, P.. Persistence in infinite-dimensional systems. SIAM J. Math. Anal. 20 (1989), 388–95.CrossRefGoogle Scholar
17Hess, P.. Periodic–parabolic boundary value problems and positivity, Pitman Research Notes in Mathematics 247 (Harlow: Longman Scientific & Technical, 1991).Google Scholar
18Hollis, S. L., Martin, R. H. Jr. and Pierre, M.. Global existence and boundedness in reaction–diffusion systems. SIAM J. Math. Anal. 18 (1987), 744–61.CrossRefGoogle Scholar
19Hutson, V. and Schmitt, K.. Permanence and the dynamics of biological systems. Math. Biosci. 111 (1992), 1–71.CrossRefGoogle ScholarPubMed
20Kirlinger, G.. Permanence of some ecological systems with several predator and one prey species. J. Math. Biol. 26 (1988), 217–32.CrossRefGoogle Scholar
21Mlak, W.. Differential inequalities of parabolic type. Ann. Polon. Math. 3 (1957), 349–54.CrossRefGoogle Scholar
22Morgan, J.. Global existence for semi-linear parabolic systems. SIAM J. Math. Anal. 20 (1989), 1128–44.CrossRefGoogle Scholar
23Pao, C. V.. Coexistence and stability of a competition–diffusion system in population dynamics. J. Math. Anal. Appl. 83 (1981), 54–76.CrossRefGoogle Scholar
24Protter, H. H. and Weinberger, H. F.. Maximum principles in differential equations (Englewood Cliffs, NJ: Prentice-Hall, 1967).Google Scholar
25Schoener, T. W.. Population growth regulated by intra-specific competition for energy or time: some simple representations. Theoret. Population Biol. 4 (1973), 56–84.CrossRefGoogle ScholarPubMed
26Schoener, T. W.. Competition and the form of habitat shift. Theoret. Population Biol. 6 (1974), 265–307.CrossRefGoogle ScholarPubMed
27Schoener, T. W.. Alternatives to Lotka–Volterra competition: models of intermediate complexity. Theoret. Population Biol. 10 (1976), 309–33.CrossRefGoogle ScholarPubMed
28Tineo, A. and Alvarez, C.. A different consideration about the globally asymptotically stable solution of the periodic n-species problem. J. Math. Anal. Appl. 159 (1991), 44–50.CrossRefGoogle Scholar
29Waltman, P.. A brief summary of persistence in dynamical systems. In Delay differential equations and dynamical systems, Lecture Notes in Mathematics 1475, pp. 31–40 (Berlin: Springer, 1991).CrossRefGoogle Scholar
30Zanolin, F.. Permanence and positive periodic solutions for Kolmogorov competiting species systems. Results Math. 21 (1992), 224–50.CrossRefGoogle Scholar
31Zhao, X. Q. and Sleeman, B. D.. Permanance in Kolmogorov competition models with diffusion. IMA J. Appl. Math. 15 (1993), 1–11.CrossRefGoogle Scholar